ca.analysis and odes - Projection of a gradient and the gradient of a projection

I think you need to reformulate your problem.

The standard definition of a vector field being `projectible' [eg Warner.
See $pi$-projectible ] requires, in the case of the projection $(x,y,z) to (x,y)$
it to have the form $F_1 (x,y){{partial} over {partial x}} + F_2 (x,y){{partial} over {partial y}} + F_3(x,y,z) {{partial} over {partial z }}$.

For your ``projection'' to be finite, you need your $f(x,y,z)$ to depend on $z$
in such a way that the integral is finite. As a consequence, its gradient will
have $F_1$ and $F_2$ either zero, or depending on $z$.

Combining your def. of `projection of a function' with the standard def. of
projection of a vector field you get the result that the only projected vector field
you can get is the zero vector fields.